Game theory worked example from AP Microeconomics

- [Instructor] What we have here is a free response question that you might see on an AP microeconomics type exam that deals with game theory, and it tells us Breadbasket and Quicklunch are the only two sandwich shops serving a small town. So, we're in an oligopoly situation where we only have a few firms. Each shop can choose to set a high price, or a low price, for sandwiches. The payoff matrix below shows the daily profits for each combination of prices that the two shops could choose. The first entry shows Breadbasket's profits, and the second entry shows Quicklunch's profits. Assuming that both shops know the information shown in the matrix, answer the following. So, just to make sure I understand what's going on here, this is saying that for example if bread, Breadbasket can either choose to charge high prices or low prices. Quicklunch can either choose to charge high prices or low prices. If Breadbasket chooses high prices and Quicklunch also chooses high prices, then what this tells us is the first one is Breadbasket's profit per day would be $105, while Quicklunch's profit per day would be $110, and then this is a situation where Breadbasket is low and Quicklunch is high price, and then Breadbasket would make 120, and Quicklunch would make 80, and then when Quicklunch has low prices we can see what are the profit of either when Breadbasket charges high prices or Breadbasket charges low. So, with that out of the way, let's try to answer these questions. Does each shop have a dominant strategy to set a high price, a dominant strategy to set a low price, or does it have no dominant strategy? So, pause the video and try to figure that out, and just as a bit of a hint, or a reminder, a dominant strategy is a strategy of regardless of what the other player does you would still be better off to make that choice. So, a dominant strategy of setting a high price would be regardless of whether the other player decides to set a high or low price a high price would always make sense for you. So, pause the video and try to see if you can answer that. All right, now let's see what Breadbasket's situation is here. So, if we think, Breadbasket of course can either choose to go high or low. Now, if Quicklunch goes high, what should Breadbasket do? Well, then we are in this column over here, and these two numbers are for Breadbasket, and so if Quicklunch goes high then Breadbasket could go high and charge 105, or it could go low and charge 120. So, if Quicklunch goes high, Breadbasket should go low. So, let me write this, Quicklunch, Quicklunch, if it chooses to go high, then Breadbasket should go low. Now, what happens if Quicklunch chooses to go low? Well, if Quicklunch chooses to go low, the two options for Breadbasket are either $40 of profit per day if they have a high price, because a lot of their business would go to Quicklunch in that situation, or they could go low price and make $75 per day. So, even if Quicklunch goes low it still makes sense for Breadbasket to go low. So, Breadbasket goes low. So, we see that Breadbasket, no matter what Quicklunch does, it makes sense for Breadbasket to go low. So, Breadbasket has a dominant strategy to set a low price. So, the dominant, dominant strategy to set, let me write that a little bit clearer, set low price. All right, now, Quicklunch, so, we could do the same type of analysis based on what Breadbasket might choose to do. So, if we see, okay, if Breadbasket goes high, what should Quicklunch do? Well, let's see, then Quicklunch should, so, if Breadbasket goes high, Quicklunch, should it go high or low? So, it would have the option of $110, or it would have an option of $130. So, if Breadbasket goes high, Quicklunch is better off going low. So, let me write that down. Quicklunch should go low. And if Breadbasket goes low what should Quicklunch do? So, if Breadbasket goes low, Quicklunch can either make 80 if it goes high or 70 if it goes low. So, in this situation if Breadbasket goes low it makes sense actually for Quicklunch to go high and make the 80. So, Quicklunch would go high. So, Quicklunch has no dominant strategy. It doesn't make sense for them to always go low, or always go high regardless of what Breadbasket is doing. Depending on Breadbasket, it might make sense for them to go low if Breadbasket goes high, or high if Breadbasket goes low. So, no dominant, dominant strategy. If the two shops do not cooperate on setting prices, what will be the profit for each shop? Well, actually, just pause the video. See if you can answer that before we work through it. Well, we already know that Breadbasket has a dominant strategy to set a low price. So, Breadbasket is gonna go low, regardless. So, that we're going to end up in this row right over here, and if we are in that situation where Breadbasket goes low regardless we know that it makes sense for Quicklunch to go high, because they could either make 80 or 70. Well, they're gonna go high, and so we're going to end up in this column right over here. So, if the two shops do not cooperate on setting prices, the profit of each shop would be Breadbasket would be making $120 a day and Quicklunch would be making $80 a day. And this is a Nash equilibrium, because on its own no firm can change its decision to optimize its prices more, because if you are in this, if you are in this cell right over here, Quicklunch can't change Breadbasket's decision. So, Quicklunch could say I'm either gonna make 80 or I'm gonna make 70, so, it wouldn't make any sense for them to switch away, and then Breadbasket, they can't make the decision for Quicklunch. They can say, hey, we're either gonna make 120 or we're gonna make 105. And so, since we can't change what Quicklunch is doing, well, of course we're going to choose to make 120. So, people will stay in this bottom left cell (mumbles) a Nash equilibrium. All right, so the next part they ask us, or they tell us, the town government is concerned that food prices are too high. It decides to give a daily subsidy of $20 to any shop that chooses to set a low price for its food items. Redraw the payoff matrix under the government subsidy system. So, like always, pause this video, and go through that exercise. It'll be interesting. All right, now, let's do this together. So, we have Breadbasket, and I'll try to write bread, actually, let me draw my little matrix first. So, it's a two by two. So, almost done with my matrix. And one more. That wasn't near the middle. There you go, right around there. Still, maybe there. And then we have that there. And then we have high, low, high, low, and then you have Quicklunch. If I were actually doing this on the test maybe I'd write this a little bit lower so it's a little bit neater, and then this is Breadbasket. And let's see. It's giving a daily subsidy of $20. So, if we could just do that, that adds to the profit of a firm that is selling at low prices. So, if they're both at high prices, that's not, they're not going to be able to get that subsidy from the government. So, we're still going to be at 105 and 110 for Breadbasket and Quicklunch respectively. Now, if Breadbasket stays high in this situation, well then, they're still only going to make $40. But in this situation right over here Quicklunch is going to get a $20 subsidy because they are choosing to go low. So, they're going to make 130 plus 20. So, I'll write that in a different color. So, I will put that in and I'll put this in red. So, 130 plus 20. They're going to make $150 because they're choosing to go low price here. That's what qualifies you for the subsidy, and in this situation if Breadbasket goes low price they're going to make 120, what they would have made in that situation, plus 20. So, this is going to be 140. But this is a situation where Quicklunch is charging a high price. So, they're not going to get the subsidy. So they're still going to make $80. And then this cell is both of them charging a low price. So they're both gonna get $20 more than what you see right over here. So, Breadbasket would make $95, $95, and Quicklunch is going to make $90 in this situation. And let's see, they ask us more questions. They say using your redrawn payoff matrix answer each of the following. Would Quicklunch choose to set a high price or a low price? Explain using specific values from your redrawn matrix. So, pause the video and see if you can answer that. All right, so we're going to look at Quicklunch now, and we could do the same type of analysis to see if they have a dominant strategy. We could say, okay, if Breadbasket goes high, goes high, what is Quicklunch going to do? So, if Breadbasket goes high, now in this situation Quicklunch will either go high and make $110 or it could go low and make 150. So Quicklunch is going to go low in this situation, and if Breadbasket goes low Quicklunch canmake $80 if it goes high or it could make $90 if it goes low. So once again, Quicklunch is going to go low. So, Quicklunch, Quicklunch now has dominant, dominant strategy for low price, for low price. And I could try to explain this out. If Breadbasket goes high, I'll write it out here. Normally you have a little book to write this in, but, if bread, I'll abbreviate, Breadbasket goes high Quicklunch is rational to make $150 instead of 110, would want to make $150 versus 110 by going low. If Breadbasket goes low, goes low, Quicklunch makes $90 versus, versus $80 by going low. All right, would the profits for Breadbasket increase, decrease, or stay the same? Explain with a comparison to your answer in part B one. Use specific values. So, before subsidy, actually pause this video and try to do this as well. Before subsidy, oh, now we'll do it together. Before subsidy, Quicklunch, or I should say Breadbasket, Breadbasket was making, and we have it up here. So, if the two shops do not cooperate on setting prices, what will be the profit of each shop? We saw Breadbasket would make $120, 'cause that was that Nash equilibrium state. After subsidy, after subsidy, so, let's look at it. We still have, Breadbasket still has a dominant strategy to go low, because if Quicklunch goes high Breadbasket would wanna make the 140 instead of 105, and if Quicklunch goes low Breadbasket would wanna make 95 instead of 40. But now they both have a dominant strategy for going low. So, we're going to end up in this bottom right, this bottom right cell. After subsidy, both have dominant, dominant strategy to go low, resulting in $95 profit for Breadbasket. And so, $95 is less than $120. So, Breadbasket's profit goes, or I should say it decreases, decreases, and we are done.